Power-law sensitivity to initial conditions within a logisticlike family of maps: Fractality and nonextensivity

Abstract

Power-law sensitivity to initial conditions, characterizing the behavior of dynamical systems at their critical points (where the standard Liapunov exponent vanishes), is studied in connection with the family of nonlinear one-dimensional logisticlike maps ${x}_{t+1}=1\ensuremath{-}a|{x}_{t}{|}^{z}$ ($z>1;$ $0<a<~2;$ $t=0,1,2,\dots{}$). The main ingredient of our approach is the generalized deviation law $limop{}_{\ensuremath{\Delta}x(0)\ensuremath{\rightarrow}0}[\ensuremath{\Delta}x(t)/\ensuremath{\Delta}x(0)]=[1+(1\ensuremath{-}q){\ensuremath{\lambda}}_{q}t{]}^{1/(1\ensuremath{-}q)}$ (equal to ${e}^{{\ensuremath{\lambda}}_{1}t}$ for $q=1,$ and proportional, for large $t,$ to ${t}^{1/(1\ensuremath{-}q)}$ for $q\ensuremath{\ne}1;$ $q\ensuremath{\in}R$ is the entropic index appearing in the recently introduced nonextensive generalized statistics). The relation between the parameter $q$ and the fractal dimension ${d}_{f}$ of the onset-to-chaos attractor is revealed: $q$ appears to monotonically decrease from 1 (Boltzmann-Gibbs, extensive, limit) to $\ensuremath{-}\ensuremath{\infty}$ when ${d}_{f}$ varies from 1 (nonfractal, ergodiclike, limit) to zero.